recognition_structure_nonempty

formal source

  65  has_self_recognition : ∃ x, recognizes x x
  66
  67/-- Any recognition structure implies nonemptiness. -/
  68theorem recognition_structure_nonempty {X : Type*}
  69    (R : RecognitionStructure X) : Nonempty X :=
  70  MetaPrinciple X ⟨R.recognizes, R.has_self_recognition⟩
  71
  72/-! ## From Recognition to Ledger -/
  73
  74/-- A minimal ledger: balanced debits and credits. -/
  75structure MinimalLedger (X : Type*) where
  76  /-- The charge of an element. -/
  77  charge : X → ℤ
  78  /-- Conservation: sum of charges is zero in any valid transaction. -/
  79  conserved : ∀ (txn : List X), (txn.map charge).sum = 0
  80
  81/-- Hypothesis class: MP forces ledger structure.
  82
  83This is a PHYSICAL HYPOTHESIS, not a mathematical theorem.
  84It captures the claim that recognition pairing forces conservation.
  85-/
  86class MPForcesLedger (X : Type*) where
  87  /-- Recognition structure exists. -/
  88  recognition : RecognitionStructure X
  89  /-- Charge assignment exists. -/
  90  charge : X → ℤ
  91  /-- Non-trivial transactions balance. -/
  92  balanced : ∀ (txn : List X), txn.length > 1 → (txn.map charge).sum = 0
  93
  94/-- The trivial model satisfies MPForcesLedger (with zero charge). -/
  95instance : MPForcesLedger Unit := {
  96  recognition := { recognizes := fun _ _ => True, has_self_recognition := ⟨(), trivial⟩ },
  97  charge := fun _ => 0,
  98  balanced := fun _ _ => by simp